Approximation Schemes for Packing Splittable Items with Cardinality Constraints

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We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of n items must be packed into as few bins as possible. Items may be split, but each bin may contain at most k (parts of) items, where k is some given parameter. Complicating the problem further is the fact that items may be larger than 1, which is the size of a bin. The problem is known to be strongly NP-hard for any fixed value of k. We essentially close this problem by providing an efficient polynomial-time approximation scheme (EPTAS) for most of its versions. Namely, we present an efficient polynomial time approximation scheme for k=o(n). A PTAS for k=Θ(n) does not exist unless P = NP. Additionally, we present dual approximation schemes for k=2 and for constant values of k. Thus we show that for any e>0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1+e.